Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information....

The following is a graph of the height of a water wave as a function of
position, at a certain moment in time.

Trace this graph onto another piece of paper, and then sketch below it the
corresponding graphs that would be obtained if

(a) the amplitude and frequency were doubled while the velocity remained
the same;

(b) the frequency and velocity were both doubled while the amplitude
remained unchanged;

(c) the wavelength and amplitude were reduced by a factor of three while
the velocity was doubled.

[Problem by Arnold Arons.]

2

(a) The graph shows the height of a water wave pulse as a function of
position. Draw a graph of height as a function of time for a specific point
on the water. Assume the pulse is traveling to the right.

(b) Repeat part a, but assume the pulse is traveling to the left.

(c) Now assume the original graph was of height as a function of time, and
draw a graph of height as a function of position, assuming the pulse is
traveling to the right.

(d) Repeat part c, but assume the pulse is traveling to the left.

[Problem by Arnold Arons.]

3

The figure shows one wavelength of a steady sinusoidal wave traveling
to the right along a string. Define a coordinate system in which the
positive x axis points to the right and the positive y axis up, such that the
flattened string would have y=0. Copy the figure, and label with "y=0" all
the appropriate parts of the string. Similarly, label with "v=0" all parts of
the string whose velocities are zero, and with "a=0" all parts whose accelerations
are zero. There is more than one point whose velocity is of the
greatest magnitude. Pick one of these, and indicate the direction of its
velocity vector. Do the same for a point having the maximum magnitude
of acceleration.

[Problem by Arnold Arons.]

4

Find an equation for the relationship between the Doppler-shifted
frequency of a wave and the frequency of the original wave, for the case of
a stationary observer and a source moving directly toward or away from
the observer.

5

Suggest a quantitative experiment to look for any deviation from the
principle of superposition for surface waves in water. Make it simple and
practical.

6

The musical note middle C has a frequency of 262 Hz. What are its
period and wavelength?

√

7

Singing that is off-pitch by more than about 1% sounds bad. How
fast would a singer have to be moving relative to a the rest of a band to
make this much of a change in pitch due to the Doppler effect?

√

8

In section 3.2, we saw that the speed of waves on a string depends on
the ratio of T/μ, i.e., the speed of the wave is greater if the string is under
more tension, and less if it has more inertia. This is true in general: the
speed of a mechanical wave always depends on the medium's inertia in
relation to the restoring force (tension, stiffness, resistance to compression,...)
Based on these ideas, explain why the speed of sound in a gas
depends strongly on temperature, while the speed of sounds in liquids and
solids does not.